Here are some of the deeper meanings that I've learned about. In summary, what is a Bayit? Besides meaning a a family as mentioned in the other answer, Bayit is a House (and is built of Stones and has a Foundation - Tzaddik Yesod Olam, Prov. 10:25) that receives and contains the Truth – and Yaakob is the aspect of Emet, truth, as is known and as explained, see below; and Bayit is something that needs a Rosh Bayit, a righteous, rightful and true Head and Leader – hence the 70 Tikunei Zohar explains BRASYT – RAS BYT (Head of the House) – and B' Reishit (Head-House – this is the gate where the righteous enter and the wicked are cast out etc.) (and many other explanations there regarding B' – Beit etc. see there).
But what about Esau's "Bayit" etc.? - Zeh l'umat zeh, as explained in Likutei Moharan #36 and elsewhere – All Yisrael are rooted in the 70 nefesh of Beit Yaakov, and contraposed to this is the seventy nations and their cravings and traits that cause prolonging of the Exile, God forbid, see there.
Likutei Moharan #36:
א. וְזֶהוּ כְּלָל
כִּי כָּל נֶפֶשׁ מִיִּשְׂרָאֵל הוּא מֻשְׁרָשׁ בְּשִׁבְעִים נֶפֶשׁ שֶׁל בֵּית יַעֲקב
וְשִׁבְעִים נֶפֶשׁ שֶׁל בֵּית יַעֲקב, מֻשְׁרָשִׁים בְּשִׁבְעִים פָּנִים שֶׁל תּוֹרָה
וְזֶה לְעֻמַּת זֶה עָשָׂה הָאֱלהִים
...
Similarly it's explained here that the souls, the "stones" as in Sefer Yetzirah – one stone builds one house, two stones build two, three build six (which is how many permutations of 3 letters) etc. - so the souls are stones that build a house and abode for Truth. But opposite to this, when enough people make an abode (they believe or they hold by something) they can make an abode for Sheker and make it a "house" too, God forbid.
Sipurei Maasiyot, Parables following the 13 Stories:
[The Two Palaces]
Know that there are two kinds of palaces, and the two palaces are identical. In one lives the king, and in the second lives a servant. And certainly in truth there is a vast difference between the palace of the king and the palace of the servant, but nevertheless it is possible to mistake one for the other, for there is a network formed by many souls binding to one another so that they become a house and palace, for one binds to another, and one to another, until they become a foundation, and then a covering, until they constitute a house and an abode. And this abode is an abode for truth, and when we need to request truth we find [it] there at that abode, that is, amidst the connection of the souls that constitute the abode for truth. And therefore the Torah has commanded, "Acharei rabim lehatoth/ Turn [judgment] to the majority [opinion]" [Ex. 23:2], for since many have bound together as one, surely the truth is there, as mentioned.
And this is the aspect of, "Kol hanefesh haba'ah leveith Yaakov/ All the souls that come to/ came to/ became the house of Yaakov" [Gen. 46:26-27 and Rashi there]. That is, the souls are what constitute the "house of Jacob," that is, an abode for truth, which is the aspect of Ya`akov, as is written, "Titen emeth leYaakov/ Grant truth to Yaakov" [Mic. 7:20]. However, be aware that diametrically opposed to this is the bond of the wicked, that many souls of the wicked bind together and become a house and abode for falsehood. And this is what the prophet warned us of, "Lo-thomrun kesher, lekhol asher-yomar ha'am hazeh kesher/ You shall not call as a band everything that this people call a band" [Isa. 8:12], for a band of the wicked is not considered, and for this it is written, "Lo thihyeh acharei rabim lera'oth/ Do not follow a multitude to do evil" [Ex. 23:2]. But behold, it is possible to mistake one of these abodes for the other, that is, between the one of truth and the one of falsehood. For falsehood imitates truth, for there too there is the binding of many souls, and a man can be mistaken and not know where the truth is nor where to drawn himself to. And know, that by means of the mitzvah of redeeming captives, one can discern between the two houses, between truth and falsehood, between the king and the servant, for falsehood is the aspect of the servant, aspect of the cursed, as in, "Arur kena'an, 'eved 'avadim/ Cursed be Canaan — a servant of servants" [Genesis 9:25].
Beit Yisrael are the aspect of כרם vineyard; שמיכה true Rabbinic ordination (also consider the aspect of the 70 Kerem and Sanhedrin, as being an aspectof the 70 souls of Yaakob aforementioned) and the writing of Yisrael, connected to Eretz Yisrael. But when God forbid unfit people are ordained then there's the opposite, and also as R' Ilash said, "women's counsel is from the out-house."
Likutei Moharan #61
3.
... וְעַל כֵּן יֵשׁ מְדִינוֹת שֶׁשִּׂכְלָם עָב וּמְגֻשָּׁם, וְכֵן יֵשׁ מְדִינוֹת שֶׁשִּׂכְלָם זַךְ וְצָלוּל, וְהַכֹּל לְפִי הַמְּדִינָה, לְפִי הַמָּזוֹן הַיּוֹצֵא מִמֶּנָּה. וְעַל כֵּן עַל־יְדֵי שֶׁנִּתְגָּרְשִׁין מִבְּחִינַת אֲוִירָא דְּאֶרֶץ־יִשְׂרָאֵל, עַל־יְדֵי בְּחִינַת הַנַּ"ל, אֲזַי אֵין יְכוֹלִין לְהִתְרַקֵּם גּוּפִים זַכִּים, וְאֵין יְכוֹלִין לְקַבֵּל נְשָׁמָה גְּדוֹלָה, וְעַל־יְדֵי־זֶה נִלְקַח סוֹד הָעִבּוּר מֵאִתָּנוּ:
Thus there are countries where the [people’s] intellect is obtuse and crass, and other countries where their intellect is clear and lucid. It all depends on the country, on the food that it produces. Therefore, when we are exiled from the air of the Land of Israel because of the aforementioned aspect, it becomes impossible to construct pure bodies, and we cannot receive a lofty soul. As a result, the Mystery of Intercalation is taken from us. {“My hand will be against the prophets who prophesy vanity and divine falsehood. They will not share in the mystery of My people, they will not be written in the writings of the House of Israel, and they will not come [back] to the land of the Israelites” (Ezekiel 13:9).}
וְזֶה שֶׁאָמְרוּ חֲכָמֵינוּ זִכְרוֹנָם לִבְרָכָה (כתובות קיב): וְהָיְתָה יָדִי אֶל הַנְּבִיאִים הַחוֹזִים שָׁוְא וְגוֹ' בְּסוֹד עַמִּי לֹא יִהְיוּ – זֶה סוֹד הָעִבּוּר; וּבִכְתַב בֵּית יִשְׂרָאֵל לֹא יִכָּתֵבוּ – זֶה סְמִיכָה; וְאֶל אַדְמַת בְּנֵי יִשְׂרָאֵל לֹא יָבוֹאוּ – כְּמַשְׁמָעוֹ. הַיְנוּ כַּנַּ"ל,
This is what the Sages, of blessed memory, said: “My hand will be against the prophets who prophesy vanity…. They will not share in the mystery of My people”—this is the Mystery of Intercalation. “They will not be written in the writings of the House of Israel”—this is ordination. “And they will not come [back] to the Land of the Israelites”—this is to be understood according to its plain meaning (Ketuvot 112a), as explained above.
בִּכְתַב יִשְׂרָאֵל לֹא יִכָּתֵבוּ זֶה סְמִיכָה; הַיְנוּ עַל־יְדֵי סְמִיכָה שֶׁמַּסְמִיכִין בְּשֵׁם רַבִּי, אֶת שֶׁאֵינָם הֲגוּנִים, עַל־יְדֵי־זֶה מַחֲלִישִׁין כְּתַב יִשְׂרָאֵל כַּנַּ"ל. וְזֶהוּ:
They will not be written in the writings of Israel—This is ordination. That is, by investing those who are unworthy with the title “rabbi,” we weaken the writings of the Jews, as explained above. And this is:
וְאֶל אַדְמַת יִשְׂרָאֵל לֹא יָבוֹאוּ, הַיְנוּ שֶׁנִּתְגָּרְשִׁין מִבְּחִינַת אֲוִירָא דְּאֶרֶץ־יִשְׂרָאֵל כַּנַּ"ל. וְזֶהוּ:
and they will not come [back] to the Land of Israel—That is, they are banished from the aspect of the air of the Land of Israel, as explained above. And this is:
בְּסוֹד עַמִּי לֹא יִהְיוּ – זֶה סוֹד הָעִבּוּר; שֶׁעַל־יְדֵי־זֶה נִלְקַח מֵאִתָּנוּ סוֹד הָעִבּוּר כַּנַּ"ל. וְכָל זֶה עַל־יְדֵי הַמּוֹתָרוֹת הַנַּ"ל:
They will not share in the mystery of My people—This is the Mystery of Intercalation. Because of this, the Mystery of Intercalation is taken from us, as explained above; all this, due to the aforementioned waste matter.
Torah 61:4
ד וְעַל־יְדֵי־זֶה אֵין לוֹ עֵצָה שְׁלֵמָה לְעוֹלָם, וְתָמִיד אֵינוֹ יָכוֹל לָתֵת עֵצָה לְנַפְשׁוֹ, וְּהוּא תָּמִיד מְסֻפָּק. כִּי כָּל עֲצוֹתָיו בִּבְחִינַת עֵצוֹת נִבְעָרוֹת, עֲצַת נָשִׁים, כִּי הַטִּנֹּפֶת שֶׁל הַמּוֹתָרוֹת עוֹלָה אֶל הַלֵּב, בִּבְחִינַת (יחזקאל י״ד:ג׳): הֶעֱלוּ גִּלּוּלֵיהֶם עַל לִבָּם; וְנַעֲשָׂה לִבּוֹ מְטֻנָּף כְּמוֹ בֵּית הַכִּסֵּא, שֶׁהוּא מְקוֹם עֲצַת הַנָּשִׁים, כְּמַאֲמַר רַב עִילָשׁ (גיטין מה): נָשֵׁי כָּל מִלֵּי מְסַדְּרִין בְּבֵית הַכִּסֵּא, שֶׁשָּׁם כָּל עֲצָתָם: 4.
- Because of this, a person never has perfect counsel and is perpetually incapable of making decisions. He is always in doubt, because all his advice is in the aspect of “foolish advice” (Isaiah 19:11), women’s counsel. This is because the filth of the waste matter rises to the heart, in the aspect of “they raised their excrement to their hearts” (Ezekiel 14:3). His heart is soiled like an outhouse, which is the place of women’s counsel. As in the statement of Rabbi Ilash: Women are wont to discuss their concerns in the outhouse (Gittin 45a), that being where they take counsel.
…
Beit Yaakob/Yisrael are alluded to as אַשְׁלו - his Eshel (pleasant garden), like a rope (also Ashla), as in "Yaakob, chevel nachalato" – due to the joy at the time of the final Redemption the joy will be so great that this rope will hang and strangle all the evil the world, and the nations will all Beit Yisrael out of exile (hence the equivalence to Yaakob here) and bring us in joy to Eretz Yisrael.
Bekhorot 8b
They [the philosophers of Athens] said to him: Build us a house in the air of the world. Rabbi Yehoshua b. Chanania uttered a Name of God and hovered between the sky and the earth. He said to them: Bring up to me bricks and mortar and I will build you a house here. They asked him: Where is the center of the world? Rabbi Yehoshua raised his finger and said to them: Here. They said to him: And who says that you are correct? He said to them: Bring ropes and measure.
They said to him: We have a pit in the field; bring it to the city. Rabbi Yehoshua said to them: Braid ropes made of bran for me and I will bring it to the city with them.
LM #24
- This is the meaning of what the Wise Men of Athens asked:
“Where is the center of the universe?” He pointed up with his finger and said, “Here!” “Who says so?” they said to him. “Bring a measuring rope and measure it,” he answered them.
They asked him: “How is it possible to grasp the Light of the Infinite One?”—for He is the center of the universe, from whom everything draws vitality and bounty.
He pointed up with his finger — This is the concept of blessings, corresponding to, “Aharon raised up his hands and blessed them,” as explained. In other words, through the blessings, the mentalities collide with the Keter and chambers are created.
Who says so? they said to him — Who is it that enters the Chambers of Exchanges, the domain of the evil forces, to elevate the holiness from there—the concept of the incense—and so that through its elevation, the blessings are elevated?
This is the meaning of, “mi yeiMaR (who says so).” Linguistically this resembles teMuRah (exchange), as in (Leviticus 27:33), “if haMeR y’MiRenu (he exchanges it)…” And he answered them:
Bring a measuring rope and measure it — In other words, the primary elevation of holiness is through joy. And in the Future, when the Jewish people will go out of the exile with joy, as in, “For you will go out with joy”—then the evil forces will be totally destroyed.
This is the meaning of “bring a measuring rope.” That is, through you and against your will, the evil forces will be eliminated. For you yourselves [the gentiles] will bring the House of Israel from the exile. They [the House of Israel] are called a “measuring rope” (אַשְׁלוּ), namely, “the rope of His inheritance” (כִּ֛י חֵ֥לֶק יְהֹוָ֖ה עַמּ֑וֹ יַעֲקֹ֖ב חֶ֥בֶל נַחֲלָתֽוֹ׃ Deuteronomy 32:9). As it is written (Isaiah 66:20), “And they will bring [all] your brethren”—the House of Israel.
With his hands, each and every gentile will bring the House of Israel from exile, by virtue of the abundant joy that will exist at that time. As is written (Psalms 126:2,3), “Then they will declare among the nations, ‘God has done greatly with these’… we were joyous.”
And through the joy, the evil forces will be eliminated, corresponding to (2 Samuel 8:2), “And they measured them with a rope, making them lie down on the ground.” This is the meaning of, “measure it”: namely, the measuring to destroy them.
מִי הוּא זֶה שֶׁיִּכְנס בְּהֵיכְלֵי הַתְּמוּרוֹת לִמְקוֹם הַקְּלִיפּוֹת
לְהַעֲלוֹת מִשָּׁם הַקְּדֻשָּׁה בִּבְחִינַת קְטרֶת כַּנַּ"ל
וְשֶׁעַל יְדֵי עֲלִיָּתוֹ יִתְעַלֶּה הַבְּרָכוֹת כַּנַּ"ל
וזה מי יימר, לשון תמורה, לשון: "אִם הָמֵר יְמִירֶנּוּ"
וְהֵשִׁיב לָהֶם, אַיְתוּ אַשְׁלֵי וּמוֹשְׁחוּ
הַיְנוּ עִקַּר עֲלִיַּת הַקְּדֻשָּׁה, עַל יְדֵי הַשִּׂמְחָה
וְלֶעָתִיד בְּצֵאת יִשְׂרָאֵל מֵהַגָּלוּת בְּשִׂמְחָה
כְּמוֹ שֶׁכָּתוּב: 'כִּי בְּשִׂמְחָה תֵצֵאוּ'
וְאָז יִכְלוּ הַקְּלִיפּוֹת לְגַמְרֵי
וְזֶה: 'אַיְתוּ אַשְׁלֵי'
הַיְנוּ עַל יְדֵיכֶם וְעַל כָּרְחֲכֶם יִתְבַּטְּלוּ הַקְּלִיפּוֹת
כִּי אַתֶּם בְּעַצְמְכֶם תָּבִיאוּ אֶת בֵּית יִשְׂרָאֵל מֵהַגָּלוּת, הַנִּקְרָאִים אַשְׁלֵי
הַיְנוּ חֶבֶל נַחֲלָתוֹ
כמו שכתוב: "וְהֵבִיאוּ אֶת אֲחֵיכֶם בֵּית יִשְׂרָאֵל"
שֶׁכָּל אֶחָד וְאֶחָד מֵהָעַכּוּ"ם
יָבִיאוּ בְּיָדָם אֶת בֵּית יִשְׂרָאֵל מֵהַגָּלוּת
עַל יְדֵי רִבּוּי הַשִּׂמְחָה שֶׁיִּהְיֶה בָּעֵת הַזּאת
כְּמוֹ שֶׁכָּתוּב: "אָז יאמְרוּ בַגּוֹיִם הִגְדִּיל ה' לַעֲשׂוֹת עִם אֵלֶּה וְכוּ', הָיִינוּ שְׂמֵחִים"
וְעַל יְדֵי הַשִּׂמְחָה יִתְבַּטְּלוּ הַקְּלִיפּוֹת
בִּבְחִינַת: 'וַיְמַדְּדֵם בַּחֶבֶל הַשְׁכֵּב אוֹתָם אַרְצָה'
וְזֶה: 'מוֹשְׁחוּ', הַיְנוּ הַמְּדִידָה
לְכַלּוֹת אוֹתָם
In Jer. 31 we also find Beit Yisrael (and Beit Yehuda) mentioned and with a mention of Boser, sour grapes – contraposed to the aforementioned Kerem, Ashel, and true Semikha.
Beit Yaakob is the aspect of the making God's greatness known in the world, even to the nations. This requires the level of Yaakob who raised it to the aspect of House/Temple higher than his fathers Yitzchak – the aspect of "praying the in the field" and Abraham – the aspect of "mountain of God" etc.
Likutei Moharan #10
ב כִּי עִקַּר גְּדֻלָּתוֹ שֶׁל הַקָּדוֹשׁ־בָּרוּךְ־הוּא הוּא, שֶׁגַּם הָעַכּוּ"ם יֵדְעוּ שֶׁיֵּשׁ אֱלֹקִים שַׁלִּיט וּמוֹשֵׁל, כַּמּוּבָא בַּזֹּהַר (יתרו דף סט:): כַּד אֲתָא יִתְרוֹ וְאָמַר: כִּי עַתָּה יָדַעְתִּי כִּי גָדוֹל ה' וְכוּ', כְּדֵין אִתְיַקַּר וְאִתְעַלָּא שְׁמָא עִלָּאָה: 2.
2. For the essence of God’s greatness is that the idolators, as well, should know there is an Almighty who rules and governs [the world]. As it is brought in the Zohar (II, 69a): When Yitro came, he said, “Now I know that God is great.” With this His Name grew greater and more exalted .
ג וְלָעַכּוּ"ם אִי אֶפְשָׁר לָהֶם לֵידַע גְּדֻלָּתוֹ שֶׁל הַקָּדוֹשׁ־בָּרוּךְ־הוּא כִּי אִם עַל־יְדֵי בְּחִינַת יַעֲקֹב, כְּמוֹ שֶׁכָּתוּב (ישעיהו ב׳:ה׳-ו׳): בֵּית יַעֲקֹב לְכוּ וְנֵלְכָה בְּאוֹר ה', כִּי הוּא גִּלָּה אֱלֹקוּתוֹ שֶׁל הַקָּדוֹשׁ־בָּרוּךְ־הוּא יוֹתֵר מִשְּׁאָר הָאָבוֹת, 3.
3. But it is impossible for the idolators to know of the Holy One’s greatness except through the aspect of Yaakov, as is written (Isaiah 2:5), “O House of Yaakov, come let us go in God’s light.” This is because [Yaakov] revealed the Holy One’s greatness even more than the other patriarchs.
כִּי אַבְרָהָם קְרָאוֹ הַר, וְיִצְחָק קְרָאוֹ שָׂדֶה (פסחים פח.), וְשָׂדֶה הוּא יוֹתֵר מֻשָּׂג וְנִצְרָךְ לְהָעוֹלָם מֵהַר. וְיַעֲקֹב קְרָאוֹ בַּיִת, שֶׁהוּא מְקוֹם יִשּׁוּב לִבְנֵי־אָדָם יוֹתֵר מִשָּׂדֶה, הַיְנוּ שֶׁיַּעֲקֹב קָרָא אֶת מְקוֹם הַבֵּית־הַמִּקְדָּשׁ, שֶׁהוּא מְקוֹם הַתְּפִלָּה, בַּיִת, שֶׁהוּא מְקוֹם יִשּׁוּב לִבְנֵי־אָדָם,
[Our Sages taught:] Avraham called it a mountain and Yitzchak called it a field (Pesachim 88a). People have a better understanding of and more need of a field than of a mountain. Yaakov, on the other hand, referred to it as a house (ibid.), which is even more suitable for human habitation than a field. For the place of the Holy Temple, the place of prayer, Yaakov called a house—a place which people inhabit.
כִּי הֶעֱלָה אֶת הַתְּפִלָּה מֵהַר וְשָׂדֶה לִבְחִינַת בַּיִת, שֶׁיֵּשׁ בּוֹ תְּפִיסָה לִבְנֵי־אָדָם יוֹתֵר מֵהַר וְשָׂדֶה, כִּי בִּבְחִינַת בַּיִת יֵשׁ גַּם לְעַכּוּ"ם הַשָּׂגָה, כְּמוֹ שֶׁכָּתוּב (ישעיהו נ״ו:ז׳): כִּי בֵיתִי בֵּית תְּפִלָּה יִקָּרֵא לְכָל הָעַמִּים, וּכְשֶׁהוּא בִּבְחִינַת בַּיִת, כְּדֵין אִתְיַקַּר שְׁמָא עִלָּאָה כַּנַּ"ל:
[Yaakov] elevated prayer from mountain and field to the aspect of house, which is more comprehensible to people than either mountain or field. For in the concept of house, idolators, too, have an understanding, as in (Isaiah 56:7), “for My house shall be called a house of prayer for all the nations.” And when it is on the level of house, then—“with this His exalted name grew still greater.”
וְזֶה פֵּרוּשׁ (תהילים מ״ח:ב׳): גָּדוֹל ה' וּמְהֻלָּל מְאֹד, כְּלוֹמַר, אֵימָתַי גָּדוֹל ה', כְּשֶׁהוּא מְהֻלָּל מְאֹד – מִסִּטְרָא דְּמוֹתָא, שֶׁהוּא בְּחִינַת עַכּוּ"ם (כְּמוֹ שֶׁאָמְרוּ רַבּוֹתֵינוּ זִכְרוֹנָם לִבְרָכָה (ב"ר פ' ט) עַל פָּסוּק (בראשית א׳:ל״א): וְהִנֵּה טוֹב מְאֹד, מְאֹד – זֶה מַלְאַךְ הַמָּוֶת), כְּשֶׁהוּא מְהֻלָּל מֵהֶם, אֲזַי הוּא גָּדוֹל, כִּי הוּא עִקַּר גְּדֻלָּתוֹ;
{“God is great, and highly praised in the city of our God, in the mountain of His holiness” (Psalms 48:2).} This is the meaning of “God is great, and m’ode (highly) praised.” That is, when is it that God is great?—when He is highly praised by the side of death, the idolators. {As our Sages taught: In the verse (Genesis 1:31), “… and behold, it was m’ode good,” the word m’ode alludes to the Angel of Death (Bereishit Rabbah 9:5).} When they praise Him, then He is great; this being the quintessence of His greatness.
וְאֵימָתַי הוּא מְהֻלָּל מֵהֶם, בְּעִיר אֱלֹקֵינוּ הַר קָדְשׁוֹ, דְּהַיְנוּ כְּשֶׁבְּחִינַת הַר נַעֲשֶׂה עִיר אֱלֹקֵינוּ, שֶׁהוּא יִשּׁוּב בְּנֵי־אָדָם, בְּחִינַת בַּיִת, שֶׁהוּא מֻשָּׂג יוֹתֵר מֵהַר וְשָׂדֶה, דְּהַיְנוּ כְּשֶׁמַּעֲלִין אֶת בְּחִינַת הַתְּפִלָּה מִבְּחִינַת הַר לִבְחִינַת עִיר וּבַיִת, שֶׁאָז יֵשׁ גַּם לָעַכּוּ"ם הַשָּׂגָה כַּנַּ"ל, אָז דַּיְקָא גָּדוֹל ה', כִּי זֶה עִקַּר גְּדֻלָּתוֹ יִתְבָּרַךְ, כְּשֶׁגַּם הָרְחוֹקִים יוֹדְעִים מִמֶּנּוּ יִתְבָּרַךְ כַּנַּ"ל:
And when is it that they praise Him? “… in the city of our God, in the mountain of His holiness.” [The idolators will praise God] when mountain becomes “the city of our God,” a place of habitation/house, which is more understood than a mountain or a field. In other words, when prayer is elevated from the level of mountain to that of city and house—so that the idolators will also gain understanding—precisely then, “God is great….” For this is the essence of the Holy One’s greatness: when those who are distant also know of Him.
ד וְעִנְיָן זֶה – לְהַעֲלוֹת הַתְּפִלָּה מִבְּחִינַת הַר וְשָׂדֶה לִבְחִינַת בַּיִת, בְּחִינַת עִיר אֱלֹקֵינוּ, כְּדֵי שֶׁיִּתְגַּלֶּה מַלְכוּתוֹ גַּם לָעַכּוּ"ם, שֶׁיִּהְיֶה לָהֶם גַּם כֵּן הַשָּׂגָה בֶּאֱלֹקוּתוֹ יִתְבָּרַךְ שְׁמוֹ – אִי אֶפְשָׁר לְהֵעָשׂוֹת כִּי אִם עַל־יְדֵי צַדִּיקֵי הַדּוֹר, כְּמוֹ שֶׁאָמְרוּ חֲכָמֵינוּ זִכְרוֹנָם לִבְרָכָה (ב"ב קטז.): מִי שֶׁיֵּשׁ לוֹ חוֹלֶה בְּתוֹךְ בֵּיתוֹ, יֵלֵךְ אֵצֶל חָכָם וִיבַקֵּשׁ עָלָיו רַחֲמִים, 4.
- And this matter of elevating prayer from the levels of mountain and field to that of house and “city of our God,” so that His sovereignty will also be revealed to the idolators in order for them to have an understanding of His Godliness, can only be achieved through the tzaddikim of the generation. As our Sages taught: When a person has someone sick at home, let him turn to a wise man who will arouse mercy for him (Bava Batra 116a).
….
Again Beit Yaakob is the aspect of revelation of God's greatness, and the Temple and Torah.
Rashi on Isa. 2.
2:3
. לפי שהוא קראו בית אל לפיכך יקרא על שמו אבל אברהם קראו בהר ה' יראה יצחק קראו שדה לשוח בשדה: to the house of the God of Jacob
Since he [Jacob] called it [the Temple site] Beth El [the house of God], therefore, it will be called on his name, but Abraham called it a mount, “On God’s mount shall He appear” (Gen. 22:14); Isaac called it a field, “to meditate in the field” (ibid. 24:63).
2:5
בית יעקב וגו'. האומות יאמרו להם כן והמקרא הזה הוא מוסב על והלכו עמים רבים וגו': O house of Jacob... The nations will say this to them, and this verse refers back to the verse [commencing with], “And many peoples shall go...”
לכו ונלכה באור ה'. יאמרו להם כן וי"ת יימרון דבית יעקב אתו וגו': come and let us go in the light of the Lord They will say this to them. Jonathan, however, renders: And those of the house of Jacob will say, [“Come and let us go in the teaching of the Law of the Lord”].
Some additional aspects:
IIRC, Akeret haBayit, the main post of the house is Rachel, the aspect of Prayer. Leah (Torah) - Ohel.
''Even maasu habonim'' - the stone the builders rejected is become the head cornerstone - David, Malkhut, speech and faith.
